Volume 3 of the Fulldome Curriculum includes a lesson based on the Titius-Bode “Rule.” In this new teaching module we present the orbits predicted by the Titius-Bode relation in a historical timeline compared to the actual planetary orbits to show students why this apparent rule was important in 18th and 19th century astronomy.

The Titius-Bode “Rule” purports to describe an apparent mathematical correspondence in the sizes of the orbits of the classic planets in our Solar System. Although the idea of some kind of relationship had been hypothesized before Johann Daniel Titius and Johann Elert Bode, their publications in 1766 and 1772, respectively, brought this relation into the limelight of astronomical thought, and hence it is named after them.

The idea is that there is a mathematical relationship between each of the orbits of the classic planets. Usually it is presented in the following form:

d=0.4+0.3x2m

… where m = -∞, 0, 1, 2, 3,… and d is the semi major axis of the planet in astronomical units.

Historically, this relationship was believed to be revealing something intrinsic about the positioning of the planets in the Solar System, that there might have been some type of resonance phenomenon within the formation of the planets within the solar nebula. The reason for this belief came out of the astronomical discoveries which were made subsequent to its popularization in the 18th century. To see this in its historical context, let’s set up a table the way it would have been constructed in the late 1700’s:

Interesting results, but the huge gap between Mars and Jupiter posed a real problem!

Shortly after the Titius-Bode “Law” became publicized, William Herschel in 1781 discovered a new planet, Uranus! This was a paradigm changing discovery, but what was just as incredible was that its semi major axis was calculated to be 19.2 AU, nearly doubling the size of the Solar System! Just as remarkable, the next predicted semi major axis from the Titius-Bode “Law” was 19.6 AU, only 2.1% different from the measured size!

This discovery started astronomers thinking that perhaps there was more to the Titius-Bode “Law” than they once thought, that perhaps it wasn’t coincidence but was revealing a yet undiscovered physical relationship within the Solar System. Twenty years later, on the first night of the new century, 1801, Father Giuseppe Piazzi discovered a new “planet,” later named Ceres.

What was truly remarkable about this new planet was that it’s semi major axes was eventually calculated with a new mathematical method by Carl Friedrich Gauss to be 2.8 AU, nearly exactly what the Titius-Bode “Law” had predicted for a planetary body residing in the gap between Mars and Jupiter! Of course soon thereafter many more bodies were discovered to reside within the gap, and by the 1850’s these objects were renamed asteroids.

However, the belief in the Titius-Bode “Law” was gaining new proponents, since it seemed to have predicted positions in which Solar System objects were subsequently discovered! The next predicted orbit would lie at 38.8 AU, and the search was on for yet another planet! Sure enough, Neptune was discovered with the aid of Newtonian physics in 1846, but its semi major axis was 30.1 AU, notthe 38.8 AU expected from the Titius-Bode relationship.

SciDome display showing the large discrepancy between Neptune’s orbit (30.1 AU) and the predicted Titius-Bode orbit of 38.8 AU

This large discrepancy led to the virtual abandonment of the Titius-Bode relationship as a physical law. However, it’s interesting to note that when Pluto was discovered in 1930 its semi major axis was determined to be 39.5 AU, very close to the previously expected distance. Of course Pluto has now been relegated to dwarf planet status because of the myriad of new objects which have been discovered in the Kuiper Belt.

The next expected semi major axis from the Titius-Bode relationship is 77.2 AU. And isn’t it interesting that Sedna’s perihelion distance is 76.1 AU, although its semi major axis is a whopping 506.8 AU!

The moral of the story seems to be that although the Titius-Bode relationship has never been convincingly proven to come from physical laws, it is noteworthy historically but also serves to perhaps warn us about jumping to conclusions even though the initial evidence may seem inviting. The Titius-Bode relationship is today such a controversial topic that Icarus, the main professional journal for presenting papers on Solar System dynamics, refuses to publish any articles on the subject!

Our ancestors were highly intelligent people who devised ingenious methods to model what they perceived to be reality in the skies. Unfortunately, they came at many of these observations with deep-rooted prejudices and a priori (preconceived) beliefs which shackled their creativity.

Figure 1: Close up of the Ptolemaic system out to the Sun’s sphere

The prevalent, far-reaching belief was that the Earth was immovable and at the center of the universe. Of course we know this is preposterous (even to the point that there is no such thing as a center to the universe), it is still a useful exercise to challenge students to prove, without leaving the Earth or using satellites, that the Earth does indeed rotate and that it revolves about the Sun.

Another a priori assumption was that celestial bodies never stopped moving, as opposed to “earthly” objects which eventually came to a halt. So, when the planets periodically went back and forth in the sky, this was unacceptable and Apollonius of Perga came up with a “solution” that allowed the wanderers to be always moving without stopping by coupling two motions at once. The planets were not simply attached to a mystical sphere (“deferent”) but they were actually attached to a mini-sphere (“epicycle”) which rotated on the larger one.

Figure 2: Mercury’s retrograde path in the Ptolemaic system

In this way planets could move around the sky but intersperse that generally easterly motion with apparent backwards motion (retrograde) when the transparent epicycle carried the planet backwards. The ancients latched on to it and it was greatly preferred to having deferents slow down, stop, go backwards, stop, then resume their original direction.

My colleague David Steelman and I created a program called Epicycles for SciDome that illustrates the main characteristics of the Ptolemaic Geocentric Model. It helps students discover the systemics of the model which can only be explained as “it just has to be that way”. Whenever that is the reasoning, it signals a problem with the theory/model. This will become obvious as we go through this paper.

Let’s first take a close look at the bodies closest to the Earth in the geocentric model, as shown in Figure 1.

The Moon moves the fastest in the sky (and even changes shape!) so it was assumed to be closest to Earth. Placement of Mercury and Venus closer to the Earth than the Sun was problematic. The theory was based upon the idea that those that appeared to move the slowest must be farthest away from Earth. The problem is that the epicycle containing Mercury, the epicycle containing Venus, and the Sun all orbited around the Earth in one year! So their order was reluctantly agreed upon because Mercury moved fastest on it epicycle, Venus next fastest, and of course the Sun had no epicycle (because it never retrograded).

Figure 3: Venus and Mercury’s retrograde paths in the Ptolemaic system

The epicycle sizes are based on arbitrarily assumed distances from Earth. The angles had to match the size of the retrograde loops seen in the sky so, looking at Figure 1, Mercury’s epicycle is tiny compared to Venus’ because Mercury’s retrograde loop is about 52 degrees in extent whereas Venus’ is about 92 degrees! The fact that Venus is farther away than Mercury from the Earth in this model requires it to be considerably larger than one might expect, but these are to scale to create the properly sized retrograde patterns.

As time is progressed a trace can be turned on which shows the retrograding patterns of the planets. Figure 2 shows a close up of Mercury and Figure 3 that of Venus.

When I ask students if they see anything peculiar as time progresses, eventually someone notices that the centers of the epicycles of Mercury and Venus are exactly and always lined up with the line connecting the Earth and Sun (the Earth-Sun Line). What explanation would the ancients have given for this? “It just has to be this way for this model to work.” Red flag number 1 that there’s something wrong with this theory.

Figure 4: The planets beyond the Sun’s sphere

Of course we know that in the Copernican heliocentric model we don’t need epicycles to cause Mercury and Venus to wobble back and forth around the Sun because they are simply closer to the Sun than Earth and they orbit the Sun. In fact, Copernicus was the first to completely untangle the motions of Mercury and Venus from the Sun’s motion.

This confusion is one rarely-discussed reason why the Copernican heliocentric model was so appealing. It unambiguously separated the motions of Mercury and Venus and even established, for the first time, their orbital periods around the Sun (88 days and 225 days, respectively).

Now observe the planets beyond the Sun, as shown in Figure 4. As we advance time another strange systematic displays itself, although this one is a lot more challenging to pick out. The Earth-Sun Line is always parallel to the planet’s epicycle radius! You can easily see this in Figure 4 now that you know to look for it.

Again, the ancients noted this “coincidence” but could never explain it other than “it has to be this way for the model to work.” Another red flag has raised itself in the flawed Ptolemaic model! The basic reason for this “coincidence” is because the retrograde motion of each planet is a function of its position relative to the Earth in its own orbit. Since we’re locking down the Earth and moving the Sun, it’s the orientation of the Earth-Sun Line that is the determining factor as to when planets exhibit their retrograde motions.

Figure 5 – The retrograde paths of the planets beyond the Sun’s sphere

When the planets leave breadcrumbs (see Figure 5) their retrograding paths become obvious. Again, the model has been carefully defined to accurately recreate the width of the retrograde loops as well as their frequency.

This is a fun and thought provoking lesson for my students because it demonstrates how intelligent and clever the ancients were in mimicking celestial motions, but it also shows how preconceived notions can weigh one down and severely complicate the model. It also clearly points out that when certain “features” of a model have no other explanation than “it has to be that way for the model to work” that the model is most likely flawed or incorrect at its core. But having the Earth move was a huge paradigm shift, and it took over 1500 years to overthrow it!

Figure 1: Page from the original printing of Sidereus Nunicius showing Galileo’s sketches of the Medicean Moons

In many of our astronomy classes, we discuss the importance of Galileo’s first telescopic observations in eventually overthrowing the Ptolemaic geocentric system. His first observations were relayed to the public in his short book Sidereus Nuncius, which is Latin for The Starry Messenger (or arguably, The Starry Message). In it he relates his observations of the Moon, the myriad of new stars he observed (with sketches of the Pleiades and Praesepe regions), and the Moons of Jupiter.

He originally called these the Medicean Stars, a call out to his potential benefactors, the four Medici brothers (the book itself was dedicated to one of them who had been a former pupil). Seeking for funds for your science… things really haven’t changed very much in 400 years…

With Starry Night, SciDome can easily reproduce the date and situations of Galileo’s observations. Others have done this in the past, and I refer you to the excellent article by Enrico Bernieri called “Learning from Galileo’s Errors” published in the Journal of the British Astronomical Association, 122, 3 (2012) which goes through his observations in detail and discusses the errors which Galileo made.

With the incredible talent of Steve Sanders (Eastern University Observatory Administrator), I have created a minilesson for Volume 3 of the Fulldome Curriculum which reproduces all of Galileo’s published observations of the Medicean moons.

Figure 2: SciDome presentation of Galileo’s first Jupiter observations from January 7, 1610 from Padua, Italy.

Using Padua, Italy as our observation location and the approximate times given for each observation in Sidereus Nuncius, we begin with the close-up view of Jupiter on the dome as seen on January 7, 1610 at approximately 6 PM local time. Next we place a slide of the view as drawn by Galileo in Sidereus Nuncius below the view to show just how accurate Galileo was in his sketches. The labels of the Galilean moons are then displayed.

Note that although all four moons presented themselves, Io and Europa were too close together to be resolved by Galileo’s homemade 20X telescope which suffered also from chromatic and spherical aberration. This is an important fact to remember, because essentially all of the “errors” which we will find in comparing his sketches to the actual viewing circumstances were because of his lack of resolution.

We proceed by advancing time in Starry Night so that the audience can watch the dance of the moons around Jupiter and stop at the next observations of Jupiter as recorded by Galileo, on January 8, 1610. Then his sketch of this configuration is displayed, and again we note the accuracy of his rough sketches.

The minilesson continues in this fashion, showing the moons moving from date to date and then presenting 21 successive sketches by Galileo as presented in Sidereus Nuncius. Galileo concluded after four nights of observations that these tiny “stars” were indeed most likely satellites of Jupiter, which was of momentous importance because it was the first time that moons had been discovered around another body.

It also indicated that a planet could move and moons “stay up with it” despite its motion, an Aristotelian argument once offered to discount that the Earth could be moving because, if it did, how could the Moon know enough to keep up with it? Obviously Jupiter had at least four moons and they had no problem staying with it!

I have found that going through many of these configurations with my students greatly enhances their appreciation of Galileo and the great discoveries that he made despite the limitations of his equipment. Presenting this minilesson engages students in the realization that Galileo was both an excellent and honest observer as well as a genius. His observations helped to lead to the downfall of the geocentric universe and the eventual acceptance of the heliocentric model